Uniform distribution of orbits of lattices on spaces of frames

Abstract

We study distribution of orbits of a lattice $\Gamma\subseteq\SL(n,\mathbb{R})$ in the the space $\mathcal{V}_{n,l}$ of $l$-frames in $\mathbb{R}^n$ ($1\le l\le n-1$). Examples of dense $\Gamma$-orbits are known from the work of Dani, Raghavan, and Veech. We show that dense orbits of $\Gamma$ are uniformly distributed in $\mathcal{V}_{n,l}$ with respect to an explicitly described measure. We also establish an analogous result for lattices in ${\rm Sp}(n,\mathbb{R})$ which act on the space of isotropic $n$-frames.